Lesson video

Everyone, my name is Ms. Coo, and I'm really happy you've chosen to learn with me today.

In today's lesson, it might be easy in some places and tricky in others, but I am here to help.

You will come across some new keywords, and maybe some keywords you've already come across before.

I hope you enjoy the lesson, so let's make a start.

In today's lesson, from the unit comparing and ordering fractions and decimals, with positive and negative numbers.

We'll be looking at limitations of technology in calculations, and by the end of the lesson, you'll be able to enter fractions as divisions on a calculator, and other technology, as well as understand the limitations of the decimal representation that results.

So let's have a look at some keywords.

First of all, a terminating decimal is one that has a finite number of digits after the decimal point.

For example, 92.

2 is a terminating decimal, as we only have one decimal place after the decimal point.

193.

3894 is a terminating decimal, as we have four digits after that decimal point.

A non example is 1.

9 with that little dot above it, and it means 1.

9 recurring, which is the same as 1.

9999 going on forever.

Another non example is pi.

Pi is a great non example.

If you put in pi into your calculator, it will only give you pi to 10 decimal places.

So you might think that pi is a terminating decimal, when it's not.

Pi has an infinite number of decimal places, it's just your calculator has only displayed around about 10 decimal places.

Now let's have a look at some other keywords, such as recurring decimals.

Now, a recurring decimal is one that has an infinite number of digits after the decimal point, and recurring digits after the decimal point are represented using dots.

And a dot is placed above the first digit that recurs, and the last digit that recurs.

For example, 0.

3 with a dot above the three.

That means 0.

3 three three going on forever.

Another example would be 0.

17, with the dot above the one and the seven.

indicating the one and the seven are the digits that recur.

Lastly, 0.

473 with the dot above the four, and a dot above the three, indicating the 4, 7, 3 are all recurring digits.

So now we've gone through our keywords, let's go through the parts of our lesson.

Our lesson will consist of two parts.

First of all, we'll be looking at correctly inputting into our calculator, and then we'll be looking at calculator displays and limitations.

So let's start by looking at correctly inputting into our calculator.

Scientific calculators are fantastic tools, but it is important to remember that calculators work out what is inputted.

Now the incorrect input will give the wrong answer, and it's not the fault of the calculator.

For example, if Laura wants to work out negative three squared, she puts in this into her calculator, and her calculator displays the answer to be negative nine.

Where is her error? So hopefully you spotted she should always use brackets when inputting negative numbers.

So what I'm going to do, is have a look at a quick check question.

I want you to have a think about why did the calculator work out the answer to be negative nine? Have a little think.

Laura wanted to work out the answer to negative three squared, but she inputted it in incorrectly, so it's not the fault of the calculator.

Scientific calculators have the priority of operations installed in them.

So what the calculator was do, was correctly applying the exponent first, and then making the final answer negative.

So it worked out the three squared first, which is nine, and then making the answer negative.

Thus we get the answer of negative nine.

Some scientific calculators have displays where the calculation you want to work out will look the same on the calculator display.

For example, let's say you wanted to work out 3.

5, add the cube of 27, subtract four, all over 2.

8 squared.

You can put this into your calculator, and the calculation will look exactly the same as what is on your calculator display.

This is really helpful as it does identify that you've inputted the question incorrectly.

However, if the input is incorrect, even though it may look the same on the calculator display, different buttons or functions may have been used.

For example, what if we were asked to work out two and a half as a decimal? Inputting two fraction button two, and then scrolling to the left twice, inputting that two will actually display two and a half like this.

But unfortunately it's not the same as the mixed number, two and a half.

And pressing execute then displays this, two bracket, one half, close bracket, one.

But what I want you to do is recognise, yes, it's clearly incorrect, but can you explain how the calculators work the answer out to be one? Well, the input has told the calculator to multiply two by one half, which has then calculated the answer to be one.

So now let's go back to that mixed number two and a half.

Which button should have been accessed to convert two and a half as a decimal? Hopefully you can spot the mixed number button is accessed via the shift key, and then the fraction button.

And then you'll see this displayed on your calculator screen, indicating the mixed number.

Then you can insert the two and a half using the numbers and the cursor keys, which we know is five over two.

And remember, if you want to convert that improper fraction into a decimal, we simply press format, scroll down to decimal, and then press execute in order to convert it to a decimal.

Now let's have a look at another check question.

I want you to use your calculator for this question.

Aisha and Alex are working out the following calculation.

They need to work out the area of the rectangle, and the rectangle has a length of five over nine metres, and a width of one third metre.

Who is more accurate and why? See if you can give it a go, and press pause if you need more time.

Well done.

So let's see how you got on.

Well, hopefully you can spot Aisha is more accurate, because the recurring decimal is exactly represented as a fraction and it's not rounded.

Alex has rounded, and because he has rounded each fraction to two decimal places, we've lost a lot of accuracy there.

So that's why Alex is less accurate than Aisha.

Now let's move on to your task.

The task wants you to circle the correct and or accurate calculation on the calculator display for the following.

For A, which one displays the square of negative five? B, which one is summing 0.

3 recurring, 0.

3 recurring and 0.

3 recurring accurately or correctly? And for C, which one is displaying five multiplied by three quarters? See if you can give it a go, and press pause if you need.

Well done.

So let's move on to question two.

Question two shows and Andeep, John and Jacob all want to buy a cake, and they divide the amount equally between them.

Each student pays the cake shop £4.

33 each.

Explain why the cake shop manager is not going to give them the cake.

So you can give it a go, and press pause if you need more time.

Well done.

So let's see how you got on.

For question one, the square of negative five is indicated here.

Remember to use brackets when we're using those negative numbers.

For B, 0.

3 recurring, add 0.

3 recurring, add 0.

3 recurring, should have been a third and a third and a third.

Remember, you need to convert the recurring decimal into a fraction for greater accuracy.

And for C five, multiplied by three quarters, well it's this one.

Remember if we're using mixed numbers, we are accessing the mix number button.

Here, it wants us to simply multiply the integer by three quarters.

Well done, you got this one right.

For question two, we needed to explain why the cake shop manager is not gonna give them the cake.

Well, it's because three lots of £4.

33 does not sum to 13 pounds.

They've actually not paid the full price, they're 1p short.

Well done everybody.

So let's have a look at the second part of our lesson.

We'll be looking at calculator displays and limitations.

So calculators are undeniably fantastic, but it is important to remember they only have a limited number of digits on display.

For example, write all the digits displayed on the calculator screen, when converting one sixth into a decimal.

So converting one sixth into a decimal obviously shows us this on our calculator screen.

But you might notice, when we get to the 10th decimal place, we have a seven.

So does this mean one sixth is a terminating decimal, and it stops at the 10th decimal place? No, because the calculator is limited to a 12 digit display, and the final digit is rounded.

It's important to recognise one sixth is a recurring decimal, but your calculator is limited in identifying that on the calculator screen.

So let's have a look at a check question.

The calculator displays the decimal 1.

23584265712.

Which of the following could this decimal be? See if you can give it a go, and press pause if you need more time.

So hopefully you've spotted they are all possible.

So let's have a look at each one individually.

It could be that the whole sequence of digits that recur, so the 23584265712, recur, so we have another 2 3 5 8, so on and so forth.

We could also have that in recurring two, so that last digit but two could recur.

We could also have the last digit of two could have been rounded.

So therefore we could have even had, 1.

23584265711, 6, where that six is recurring, thus making that 11th decimal place rounded to two.

And it could even be a terminating decimal, where it stops at that 11th decimal place.

This is a lovely question, as it really does show the limitations of using a calculator.

It's important to remember there is no button on the Classwiz to show the recurring decimal, and we must use the fractional equivalent if known, or type in the recurring decimal to a minimum of 18 decimal places for a number less than one.

For example, if you put in 0.

16666, so on and so forth, but it's less than 18 decimal places, it will not convert it into one sixth.

The output would be in this case 833 over 5,000.

We don't want that.

We want the output to be one sixth.

So we have looked at how scientific calculators convert terminating decimals into fractions easily.

However, there is a limit at which the scientific calculator will convert the decimal into a fraction, including terminating and recurring decimals.

So let's see if you can investigate and find out when a scientific calculator stops converting a terminating decimal into a fraction.

I want us to do a little investigation.

So for a number less than one, I'm going to look at investigating the number of decimal places before the calculator converts it into a fraction.

For example, if I were to put 0.

1, which is one decimal place into our calculator, does the calculator convert into a fraction? Yes, it does.

Next two decimal places.

If I were to put in 0.

12 in, for example, to two decimal places, does it convert it into a fraction? Yes, it does.

What I want you to do, is investigate the remaining decimal places, and fill in a table to identify, when does the calculator stop converting the decimal into a fraction? See if you can give it a go, and press pause if you need more time.

So remember, you can choose any number less than one, for two, three decimal places.

I've chose 0.

123.

Yes it does convert it into a fraction.

Four decimal places, 0.

1234.

Yes it does convert it into a fraction.

Five decimal places, it converts to a fraction.

And even six decimal places, it converts it into a fraction.

But when we get to a seventh decimal place for a number less than one, it does not convert it into a fraction.

So for a number less than one, the scientific calculator stops converting the terminating decimal into a fraction, when there are seven or more decimal places.

But what about numbers greater than one? Well, let's do another investigation.

I want you to investigate how many digits the decimal must have, in order for the scientific calculator to convert the number into a fraction.

See if you can investigate this, and press pause if you need more time.

Well done.

Let's see what you found out.

Well, to convert terminating decimals to fractions, they must have less than seven digits.

For example, 12.

3456, it converts it into a fraction.

1.

23456.

It converts it into a fraction.

But when I go beyond those seven digits, when the number is greater than one, it does not convert it into a fraction.

So hopefully you can spot the limitations of when using a scientific calculator.

Now it's time for your task.

I want you to identify if the statement is true or false When using the Casio Classwiz fx 991 calculator.

To identify if the following statements are true or false.

Press pause if you need more time.

Well done.

So let's move on to question two.

Question two says Andeep has discovered a way to find the fractional form of a decimal, when the calculator does not do it automatically.

What is the fractional form of 0.

346774? And what is the simplified fractional form of 0.

346774? See if you can give it a go, and press pause if you need.

Great work.

So let's move on to question three.

Question three shows a calculator display of this number, and you're asked to give three examples in decimal form, of what this number could be if it were a recurring decimal.

For B, I want you to give an example of a fraction if it could be, if it was a terminating decimal.

This is a great question to really make you think.

Well done.

So let's go through these answers.

For question one, hopefully you've spotted the calculator will convert 0.

2673918 into a fraction automatically.

It's false because we have seven decimal places, and we know for numbers which are less than one, the calculator does not convert it to its fractional form.

The second statement says it's more accurate to use 0.

16666 rather than one sixth.

That is false.

One sixth is more accurate, 0.

1666 is actually incorrectly rounded to five decimal places.

The next statement says, inputting a recurring digit to 18 decimal places will convert it to a fraction.

That is true.

And the last statement says the priority operations is applied to all calculations.

That is also true.

Well done if you got those right.

For question two, Andeep's discovered a way to find the fractional form of a decimal, when the calculator does not do it automatically.

So what do you think the fractional form of 0.

346774 is? Now hopefully you've spotted it's going to be 346,774 over 1 million.

Now part B wants us to write the simplified fraction of 0.

346774.

Simplifying this further gives us 173,387 over 500,000.

Really well done if you got that one right.

For question two, we had to give three examples in decimal form of what this number could be if it was a recurring decimal.

Well, there are lots of different answers here.

Here's just some examples of non terminating decimals.

You could have had 0.

05395623545454 so on and so forth.

Or you could have had a recurring number of eights.

Remember the digit of the four would've been rounded up, to give you that last digit of five on your calculator display.

Finally, you could have also had a recurring digit of one.

Well done, if you got that one right.

Now we had to give an example of what it could be, if it was a terminating decimal.

Well it would be 539,562,635 over 10 billion.

So really well done if you got this one right.

So in summary, using fractions to represent terminating or non terminating decimals is generally better for accuracy.

Scientific calculators are undoubtedly fantastic tools, but only output what has been inputted.

So you must check your calculations carefully.

Scientific calculators do have their limitations, including inputting recurring decimals.

Remember, input more than 18 decimal places when the number is less than one.

When converting terminating decimals to fractions, they must have at least seven significant figures.

Also they can only display a certain number of digits on the screen, and then it'll round.

So you may not know that it's been rounded.

Huge well done today.

It was great working with you.